Equilibrium Valuation of Options on the Market Portfolio with Stochastic Volatility and Return Predictability

This paper uses an extension of the equilibrium model of Lucas (1978) to study the valuation of options on the market portfolio with return predictability, endogenous stochastic volatility and interest rates. Equilibrium conditions imply that the mean-reverting of the rate of dividend growth induces the predictable feature of the market portfolio. Although the actual drift of the price for the market portfolio does not explicitly enter into the option price formula when the equivalent martingale pricing principle is used, parameters underlying the predictable feature affect option prices through their influence on endogenized volatility and interest rates. Equilibrium conditions also review that there is strong interdependence between the equilibrium price process for the market portfolio and its volatility process, both of which are induced by the process for aggregate dividend. Closed-form pricing formulas for options on the market portfolio incorporate both stochastic volatility and stochastic interest rates. With realistic parameter values, numerical examples show that stochastic volatility and stochastic interest rates are both necessary for correcting the pricing biases generated by the Black-Scholes model. In addition, Closed-form solutions for European bond option prices are obtained, which encompass the Vasicek (1977) model and the Cox-Ingersoll-Ross (1985) model. In this sense, the current model provides a consistent way to price options written on the market portfolio and the bonds.